Antimagicness of subdivided fans

Author(s): ^¹, ^², ^¹, ^³

¹Department of Mathematics, NCBA & E, Quaid-e-Azam Campus, Township Lahore, Pakistan.

²Govt. Degree College (B), Sharqpur Sharif, Pakistan.

³Department of Mathematics, National University of Singapore, Singapore.

Copyright © Afshan Tabassum, Muhammad Awais Umar, Muzamil Perveen, Abdul Raheem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: 17/11/2019
Accepted: 17/01/2020
Published: 09/02/2020

Abstract

A graph $\Gamma$ (simple, finite, undirected) with an $\Omega$-covering has an $(\alpha,\delta)$-$\Omega$-antimagic labeling if the weights of all subgraphs $\Omega$ of graph $\Gamma$ constitute an arithmetic progression with the common difference $\delta$. Such a~graph is called super $(\alpha,\delta)$-$\Omega$-antimagic if $\nu(V(\Gamma))= \{ 1,2,3,\dots,|V(\Gamma)|\}$. In the present paper, the cycle coverings of subdivision of fan graphs has been considered and results are proved for several differences.

Keywords: $\Omega$-covering, super $(\alpha,\delta)$-$\Omega$-antimagic graph, cycle-antimagic, super cycle-antimagic, fan graphs.

1. Introduction

Let $\Gamma=(V(\Gamma),E(\Gamma))$ be a~finite simple and undirected graph with a~family of subgraphs $\Omega_1, \Omega_2, \dots, \Omega_t$ such that every element of $E(\Gamma)$ belongs to $\Omega_i\cong \Omega,\ i=1, 2, \dots, t$, then $\Gamma$ admits an~ $\Omega$-covering. An $\Omega$-covered graph $\Gamma$ with $\nu$ is called an $(\alpha, \delta)$-$\Omega$-antimagic if $wt_\nu(\Omega)=\{\alpha,\alpha+\delta,\dots,\alpha+(t-1)\delta\}$ where the associated $\Omega$-weights denoted by $wt_\nu(\Omega)$ are defined as $$wt_\nu(\Omega) = \sum\limits_{v\in V(\Omega)} \nu(v) + \sum\limits_{e\in E(\Omega)} \nu(e).$$ and $\alpha>0$ and $\delta\ge 0$ are two integers, $t$ is the number of $\Omega_i\cong \Omega$. For a total labeling $\nu$ to be super we require $\nu(V(\Gamma))= \{ 1,2,\dots,|V(\Gamma)|\}$.

The results about $\Omega$-(super)magic graphs with $\Omega$ as cycle, path and tree can be studied in [1, 2, 3, 4, 5, 6, 7].

Inayah et al. [8] introduced the $(\alpha,\delta)$-$\Omega$-antimagic labeling. We refer [9, 10, 11] for some results on super $(\alpha,\delta)$-$\Omega$-antimagic labeling. In [11], Lih proved that $F_n$ is $C_3$-supermagic for every $n$ except $n \equiv 2$ (mod 4). In [12], Ngurah et al. proved that $F_n$ is $C_3$-supermagic for every $n\ge 2$. In the present paper, we proved the super $(\alpha,\delta)$-$C_{r+2k+3}$-antimagic labelings of subdivided fans for differences $\delta=0, 1, 2, 3, 4$.

2. Preliminaries

In this section, we give basic definitions of concepts concerning a subdivided fan $F_n(r,k)$.

Definition 1. A graph $F_n\cong P_n+K_1$ is called fan graph obtained by the join of path $P_n$ and one isolated vertex $K_1$.

The central vertex, or the hub vertex is of degree $n$ and path vertices are the other ones. Spokes are the adjacent edges of central vertex and path edges are the remaining edges. \begin{eqnarray*}V(F_n)&=&\{c\}\cup \{x_1, x_2, \dots, x_{n}\},\\ E(F_n)&=& \{x_1x_2,x_2x_3, \dots,x_{n-1}x_n\}\cup\{ cx_1,cx_2,\dots,cx_{n}\}. \end{eqnarray*}
Definition 2. The subdivided fan $F_n(r,k)$ is the graph obtained from a fan $F_n$ by inserting $r \geq 1$ new vertices $\{v_1^{(i)}, \dots, v_r^{(i)}\}$ into each path edge $x_ix_{i+1}, 1\leq i\leq n-1$, denoted by $P_{x_ix_{i+1}}$-vertices and by inserting $k \geq 1$ new vertices $\{w_1^{(i)}, \dots, w_k^{(i)}\}$ into every spoke $cx_i, 1 \leq i \leq n$, denoted by $S^{(i)}$-vertices.
\begin{eqnarray*} E(P_{x_ix_{i+1}})&=&\{x_iv_1^{(i)},v_2^{(i)}v_3^{(i)}, \dots, v_{r-1}^{(i)}v_{r}^{(i)}, v_r^{(i)}x_{i+1}, 1 \leq i \leq n-1\},\\ E(S^{(i)})&=&\{cw_1^{(i)},w_2^{(i)}w_3^{(i)}, \dots, w_{k-1}^{(i)}w_{k}^{(i)}, w_k^{(i)}x_{i}, 1 \leq i \leq n\}.\end{eqnarray*} Let $C_{r+2k+3}^{(i)}$ be the $i^{\text{th}}$-subcycle. For the weight of $i^{\text{th}}$-subcycle $C_{r+2k+3}^{(i)}$, we obtain

\begin{eqnarray} wt_{\psi}(C_{r+2k+3}^{(i)})&=& \sum\limits_{u\in V(C_{r+2k+3}^{(i)})} \psi(u)+ \sum\limits_{e\in E(C_{r+2k+3}^{(i)})} \psi(e)\nonumber\\ &= &\left(\psi(x_{i})+\psi(x_{i+1})+\psi(c)+\sum\limits_{v\in V(P_{x_ix_{i+1}})} \psi(v)+ \sum\limits_{w\in V(S^{(i)})} \psi(w)\right) \nonumber\\ &&+ \left( \sum\limits_{e\in E(P_{x_ix_{i+1}})} \psi(e)+ \sum\limits_{e\in E(S^{(i)})} \psi(e)+ \sum\limits_{e\in E(S^{(i+1)})} \psi(e)\right).\label{subfan} \end{eqnarray}

(1)

where indices $i$ are taken modulo $n$.
3. Main results
In this section, we introduce the super $(\alpha,\delta)$-$C_{r+2k+3}$-antimagic labelings of subdivided fans for differences $d=0, 1, 2, 3, 4$.
Theorem 1. Let $r, k \geq 1$ and $n\ge3$ be positive integers. The subdivided fan $F_n(r,k)$ is super $(\alpha, \delta)$-$C_{r+2k+3}$-antimagic for difference $\delta=0, 1, 4$.

Proof. The total labeling $\psi_{\delta}$ is defined as: $$\psi_{\{\delta\}}(c)= 1$$ \[\psi_{\{0,4\}}(x_{i})=\begin{cases} \lceil\frac{n}{2}\rceil+2-\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ n+2-\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] \[\psi_{\{1\}}(x_{i})=\begin{cases} 1+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 1+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] \[\psi_{\{0,1\}}(cw_1^{(i)})=\begin{cases} 2(n-1)(r+1)+2(nk+1)+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2(n-1)(r+1)+2(nk+1)+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] $$\psi_{\{4\}}(cw_1^{(i)})=2n(r+k)+(3n-2r+1)-i.$$ For $\delta=0,1,4$ \begin{eqnarray*}\psi_{\delta}(V(P_{x_ix_{i+1}}))&=&\{(n-1)j+2+i:1 \leq i \leq n-1, 1 \leq j \leq r\} \\ \psi_{\delta}(E(P_{x_ix_{i+1}}))&=&\{(n-1)(2r+2-j)+n(k+1)+2-i:1 \leq j \leq r+1\}\\ \psi_{\delta}(V(S^{(i)}))&=&\{ r(n-1)+1+nj+i:1 \leq i \leq n, 1 \leq j \leq k \}\\ \psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})&=&\{2n(r+k)+(3n-2r+1)-nj-i:1 \leq j \leq k\}\end{eqnarray*} where indices $i$ are taken modulo $n$. Evidently $\psi_{\delta}$ is a super labeling as $V(F_n(rk)))=\{1, 2, \dots, n(k+r+1)-r+1\}$. The spoke vertices are labeled with the numbers $n+2,n+3 , \dots, n+2k+1$ and the path edge vertices are labeled with $n+2k+2, n+2k+3, \dots, n(k+r+1)-r+1$. Clearly,

\begin{align} \sum\limits\left(\psi_{\delta}(V(S^{(i)}))+\psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})\right)&=3rk(n-1)+k(3n+2nk+2)\nonumber\\ \sum\limits\left(\psi_{\delta}(V(P_{x_ix_{i+1}}))+ \psi_{\delta}(E(P_{x_ix_{i+1}}))\right)&=nr(2r+k+4)+n(k+2)+ +r(1-2r)+1-i\label{subfana} \end{align}

(2)

According to (1) and (2), we obtain

\begin{eqnarray} wt_{\psi_{0}}(C_{r+2k+3}^{(i)})&=&4(n-1)(r+1)+4(n k+1)+6rk(n-1)+2\left\lceil\frac{n}{2}\right\rceil+n+3+2k(3n+2nk+2) \nonumber\\ &&+nr(2r+k+4)+n(k+2)+r(1-2r)+1\nonumber\\ wt_{\psi_{0}}(C_{r+2k+3}^{(i)})&=&n+2\left\lceil\frac{n}{2}\right\rceil+nk(7r+4k+11)+2nr(r+4)+r(1-2r-6k)+2(3n+2k-2r)+4.\label{subfan0} \end{eqnarray}

(3)

Equation (3) shows that all $C_{r+2k+3}^{(i)}$-weights are independent of $i$. According to (1) and (2), we obtain

\begin{eqnarray} wt_{\psi_1}(C_{r+2k+3}^{(i)})&=&4(n-1)(r+1)+4(n k+1)+6rk(n-1)+2\left\lceil\frac{n}{2}\right\rceil+5+2i+2k(3n+2nk+2) \nonumber\\ &&+nr(2r+k+4)+n(k+2)+r(1-2r)+1-i\nonumber\\ wt_{\psi_1}(C_{r+2k+3}^{(i)})&=&2\left\lceil\frac{n}{2}\right\rceil+nk(7r+4k+11)+2nr(r+4)+r(1-2r-6k)+2(3n+2k-2r)+6+i.\label{subfan1} \end{eqnarray}

(4)

Equation (4) shows that all $C_{r+2k+3}^{(i)}$-weight consists of consecutive integers. According to (1) and (2), we obtain

\begin{eqnarray} wt_{\psi_4}(C_{r+2k+3}^{(i)})&=&4(n-1)(r+1)+4(n k+1)+6rk(n-1)+2\left\lceil\frac{n}{2}\right\rceil+5+2i+2k(3n+2nk+2) \nonumber\\ &&+nr(2r+k+4)+n(k+2)+r(1-2r)+1-i\nonumber\\ wt_{\psi_4}(C_{r+2k+3}^{(i)})&=&n+\left\lceil\frac{n}{2}\right\rceil+r(1-2r)+k(4nk-5r+9)+2n(3k+4)(r+1)-6n+7-4i.\label{subfan4} \end{eqnarray}

(5)

Equation (5) shows that all $C_{r+2k+3}^{(i)}$-weight constitute an arithmetic progression with common difference $\delta=4$. This completes the proof.

Theorem 2. Let $r, k \geq 1$ and $n\ge3$ be positive integers. The subdivided fan $F_n(r,k)$ is super $(\alpha, \delta)$-$C_{r+2k+3}$-antimagic for difference $\delta=2,3,5$.

Proof. The total labeling $\psi_{\delta}$ is defined as: \begin{align*} \psi_{\{\delta\}}(c)&= 1\\ \psi_{\{\delta\}}(x_i)&= 2i\\ \psi_{\{\delta\}}(v_r)&= 2i+1 \end{align*} $$\psi_{\{2\}}(cw_1^{(i)})=2n(r+k)+(3n-2r+1)-i$$ \[\psi_{\{3\}}(cw_1^{(i)})=\begin{cases} 2n(r+k)+(3n-2r+1)-\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2n(r+k)+(3n-2r+1)-\lceil\frac{n}{2}\rceil-\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2)\\ \end{cases}\] \[\psi_{\{5\}}(cw_1^{(i)})=\begin{cases} 2\{r(n-1)+n(k+1)\}+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2\{r(n-1)+n(k+1)\}+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2)\\ \end{cases}\] For $\delta=2,3,5$ \begin{eqnarray*}\psi_{\delta}(V(P_{x_ix_{i+1}}))&=&\{n+(n-1)j+1+i:1 \leq i \leq n-1, 1 \leq j \leq r-1\} \\ \psi_{\delta}(E(P_{x_ix_{i+1}}))&=&\{(n-1)(2r+2-j)+n(k+1)+2-i:1 \leq j \leq r+1\}\\ \psi_{\delta}(V(S^{(i)}))&=&\{ r(n-1)+1+nj+i:1 \leq i \leq n, 1 \leq j \leq k \}\\ \psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})&=&\{2n(r+k)+(3n-2r+1)-nj-i:1 \leq j \leq k\} \end{eqnarray*} where indices $i$ are taken modulo $n$. Evidently $\psi_{\delta}$ is a super labeling as $V(F_n(rk)))=\{1, 2, \dots, n(k+r+1)-r+1\}$. The spoke vertices are labeled with the numbers $n+2,n+3 , \dots, n+2k+1$ and the path edge vertices are labeled with $n+2k+2, n+2k+3, \dots, n(k+r+1)-r+1$. Clearly,

\begin{align} \sum\limits\left(\psi_{\delta}(V(S^{(i)}))+\psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})\right)&=3rk(n-1)+k(3n+2nk+2)\nonumber\\ \sum\limits\left(\psi_{\delta}(V(P_{x_ix_{i+1}}))+ \psi_{\delta}(E(P_{x_ix_{i+1}}))\right)&=n(r+1)(k+3)+2n(r^2-1) +r(n-2r+1)+1.\label{subfane} \end{align}

(6)

According to (1) and (6), we obtain

\begin{eqnarray} wt_{\psi_{2}}(C_{r+2k+3}^{(i)})&=& 2i+3+n\{k(r+5)+7r+3\}+2nk(2k+3)+2n(r^2-1)+2(3n-2r+2k+1)\nonumber\\ wt_{\psi_{2}}(C_{r+2k+3}^{(i)})&=& nk(4k+r+11)+n(2r^2+7r+1)+ 2(3n-2r+2k)+5+2i. \label{subfan2} \end{eqnarray}

(7)

Equation (7) shows that all $C_{r+2k+3}^{(i)}$-weights constitute an arithmetic progression with common difference $\delta=2$. According to (1) and (6), we obtain

\begin{eqnarray} wt_{\psi_{3}}(C_{r+2k+3}^{(i)})&=& 6n-4r+5+3i-\left\lceil\frac{n}{2}\right\rceil+r(n-2r+1)+6kr(n-1)\nonumber\\ &&+2kn(2k+3)+4k+n(2r^2+5k+7r+rk+1)\nonumber\\ wt_{\psi_{3}}(C_{r+2k+3}^{(i)})&=&2nr(r+4)+nk(4k+r+11)+6rk(n-1)- r(2r+3)+7n+4k-\left\lceil\frac{n}{2}\right\rceil+5+3i. \label{subfan3} \end{eqnarray}

(8)

Equation (8) shows that all $C_{r+2k+3}^{(i)}$-weights constitute an arithmetic progression with common difference $\delta=3$. According to (1) and (6), we obtain

\begin{eqnarray} wt_{\psi_{5}}(C_{r+2k+3}^{(i)})&=& nk(6r+11)+nr(3r+8)-r(2r+3)+2k(2nk-3r+2)+5n+\lceil\frac{n}{2}\rceil+5+5i.\label{subfan5} \end{eqnarray}

(9)

Equation (9) shows that all $C_{r+2k+3}^{(i)}$-weights constitute an arithmetic progression with common difference $\delta=5$.

Concluding remarks
An $\Omega$-covering graphs is the extension of the edge-antimagic labeling and generalizes the structure for $\Omega$-antimagic labeling. Several results concerning $\Omega$-antimagic labelings for different families of graphs are proved and available in literature. In the present manuscript, the super $(\alpha,\delta)$-$C_n$-antimagicness of subdivided fans has been considered for few of differences. One can work to extend the labeling for further differences greater than $5$.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.

References:

Gutiérrez, A., & Lladó, A. (2005). Magic coverings. Journal of Combinatorial Mathematics and Combinatorial Computing, 55, 43-56. [Google Scholor]

Inayah, N., Lvpez, A., \& Moragas, J. (2012). Magic and antimagic H-decompositions. Discrete Mathematics, 312(7), 1367-1371. [Google Scholor]

Jeyanthi, P., & Selvagopal, P. (2010). More classes of $H$-supermagic graphs. International Journal of Algorithms, Computing and Mathematics, 3(1), 93-108. [Google Scholor]

Lladó, A., & Moragas, J. (2007). Cycle-magic graphs. Discrete Mathematics, 307(23), 2925-2933.[Google Scholor]

Maryati, T. K., Baskoro, E. T., & Salman, A. N. M. (2008). $P_{h}$-supermagic labelings of some trees. Journal of Combinatorial Mathematics and Combinatorial Computing, 65, 197.[Google Scholor]

Maryati, T. K., Salman, A. N. M., & Baskoro, E. T. (2013). Supermagic coverings of the disjoint union of graphs and amalgamations. Discrete Mathematics, 313(4), 397-405.[Google Scholor]

Ngurah, A. A. G., Salman, A. N. M., & Susilowati, L. (2010). H-supermagic labelings of graphs. Discrete Mathematics, 310(8), 1293-1300.[Google Scholor]

Inayah, N., Salman, A. N. M., & Simanjuntak, R. (2009). On (a, d)-H-antimagic coverings of graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 71, 273-281. [Google Scholor]

Inayah, N., Simanjuntak, R., Salman, A. N. M., & Syuhada, K. I. A. (2013). Super (a, d)-H-antimagic total labelings for shackles of a connected graph $H$. Australasian Journal of Combinatorics, 57, 127-138. [Google Scholor]

Umar, M. A. (2019). Cyclic-antimagic construction of ladders. Engineering and Applied Science Letter, 2(2), 43-47. [Google Scholor]

Ali, N., Umar, M. A., Tabassum, N., & Raheem, L. (2018). Super $(a,d)$-C3-antimagicness of a corona graph. Open Journal of Mathematical Sciences, 2(1), 371-378. [Google Scholor]

Lih, K. W. (1983). On magic and consecutive labelings of plane graphs. Utilitas Math, 24, 165-197. [Google Scholor]

Contents

Open Journal of Mathematical Science (OMS)

Antimagicness of subdivided fans

Abstract

1. Introduction

2. Preliminaries

3. Main results

Concluding remarks

Author Contributions

Competing Interests

References:

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